Computer numerical simulation is of great importance in the analysis of non-ideal characteristics of power electronics systems. However, for power electronics systems taking account of non-ideal models of power switches and stray parameters, the simulation models have characteristics of multi-time scale, discontinuity, non-linearity, and strong stiffness. When solving such systems, traditional numerical algorithms based on the discretization of time (such as trapezoidal method, Runge-Kutta method, etc.) have the following difficulties. There is a difficulty in determining a step size. Fixed-step algorithms may lead to wrong solutions due to error accumulation; while variable-step algorithms will spend much computation time on adjusting step size, and may choose too-small time steps in order to capture the trajectories of fast state variables, which causes the simulation difficult to go forward. Theses algorithms are inefficient. Power electronics systems always contain many discrete events, which leads to discontinuities in mathematics. These algorithms, whether fixed-time step algorithms or variable-step algorithms, cannot integrate across the discontinuity points, and the iterations to find discontinuity points will consume a large amount of computation time and slow down the simulation. Discrete-time algorithms may lead to spurious oscillations or divergence in the simulation due to multi-time scale.
In order to help overcome the shortcomings of traditional discrete-time algorithms, in 2001, Ernesto Kofman et al. proposed an algorithm based on the discretization of event, named Quantized State Systems (QSS) algorithm. (See Kofman E, Junco S., Quantized-state systems: a DEVS Approach for continuous system simulation [J], Simulation Transactions of the Society for Modeling & Simulation International, 2001, 18(3):123-132, which is incorporated herein by reference in its entirety.) Different from classical discrete-time algorithms, in QSS, every state variable is quantized first by means of quantization function (Q-function), then the vector of derivatives is calculated according to the values of Q-functions rather than the values of state variables themselves, and the time which takes from one quantized state to a next quantized state can be calculated to push the simulation. QSS has some advantages over classical time-driven algorithms. QSS has excellent mathematical properties, such as good stability and convergence, because global errors in QSS algorithm can be constrained by the quantum. There is no iteration in QSS because all state variables are approximated by piecewise constant Q function. Time intervals between two points of discontinuity can be easily calculated by simple linear operations, which means higher efficiency. QSS is a variable-step method by nature. The time from one quantized state to a next quantized state is determined by the derivatives of the state variables and the relevant quantum. When the derivative is large, the time step calculated is a small value. On the contrary, a small derivative results in a large time step. Thus, no extra computation is needed for finding a suitable step. All the above advantages indicate that QSS possesses potential in the simulation of power electronics systems.
In spite of the advantages mentioned above, the efficiency and stability of the QSS algorithm when dealing with stiff systems are not satisfactory. Power electronics systems are typical stiff systems because they involve a large number of switches and multi-time scale processes. If stiff integrator algorithms are not used when solving stiff systems, the maximum step size permitted will be determined by the smallest modulus of the Jacobian matrix's eigenvalues rather than the vector of the derivatives, and the step size must be very small to satisfy the demand of stability, which will significantly slow down the speed of simulation. All discrete-time stiff integrator algorithms are implicit backward algorithms, while QSS is an explicit forward one, and QSS does not perform well in efficiency and stability when solving systems with strong stiffness.
Ernesto Kofman et al. proposed a backward QSS algorithm (BQSS). In BQSS, a group of quantized values of the state variables are selected as the vector Q for the next computation step, based on which the signs of the derivatives calculated can make every state variable approach its quantized value at the next step (See Migoni G, Kofman E, Cellier F. Quantization-based new integration methods for stiff ordinary differential equations [J], Simulation, 2012, 88(4):387-407, which is incorporated herein by reference in its entirety). As an implicit event-driven algorithm, the main problem of BQSS is how to select the vector Q correctly and efficiently. The derivative of a state is variable is not only affected by its own quantized value but also by the quantized values of other state variables. Thus, the selection of quantized values of state variables affects each other. At each computation step, there are two alternatives for the quantized value of each state variable, and thus enumeration is not feasible in high dimensional systems due to a huge number of combinations. The existing implementation schemes of BQSS are all based on enumeration, and do not take into account the interactions and constraints among quantized values. Thus, in complex systems like power electronics systems, these implementation schemes cannot find the correct vector Q, and the few existing examples are all simple low dimensional systems. There is no effective scheme for complex high dimensional systems yet.